EDGE Math 8

Laws of Exponents

Consider the expression ${(a b)}^{3} .$ Using the associative and commutative properties of multiplication, you can rewrite this expression as follows:

{(a b)}^{3} = (a b) (a b) (a b) = a b a b a b

= a a a b b b = (a a a) (b b b) .

Then using the definition of a positive integer exponent, $(a a a) (b b b)$ can be written as $a^{3} b^{3} .$

Thus, ${(a b)}^{3} = a^{3} b^{3} .$ This example illustrates the following law of exponents:

Rule for a Product Raised to a Power
If $a$ and $b$ are real numbers and $m$ is a positive integer, then

{(a b)}^{m} = a^{m} b^{m} .

This rule states that when raising a product to a power, you have to raise each factor to that power.

Example

Simplify each expression.

1. ${(a b)}^{5}$

2. ${(x y)}^{8}$

3. ${(2 p)}^{4}$

4. ${(- 3 x)}^{2}$

Solution

Use the Rule for a Product Raised to a Power.

1. ${(a b)}^{5} = a^{5} b^{5}$

2. ${(x y)}^{8} = x^{8} y^{8}$

3. ${(2 p)}^{4} = 2^{4} p^{4} = 16 p^{4}$

4. ${(- 3 x)}^{2} = {(- 3)}^{2} x^{2} = 9 x^{2}$